diff --git a/dogsdogsdogs/src/operators/half_join.rs b/dogsdogsdogs/src/operators/half_join.rs
index 338a994aa..a2920c163 100644
--- a/dogsdogsdogs/src/operators/half_join.rs
+++ b/dogsdogsdogs/src/operators/half_join.rs
@@ -186,6 +186,9 @@ where
             let timer = std::time::Instant::now();
             let mut work = 0;
 
+            // New entries to introduce to the stash after processing.
+            let mut stash_additions = HashMap::new();
+
             if let Some(ref mut trace) = arrangement_trace {
 
                 for (capability, proposals) in stash.iter_mut() {
@@ -231,6 +234,35 @@ where
                         }
 
                         proposals.retain(|ptd| !ptd.2.is_zero());
+
+                        // Determine the lower bound of remaining update times.
+                        let mut antichain = Antichain::new();
+                        for (_, initial, _) in proposals.iter() {
+                            antichain.insert(initial.clone());
+                        }
+                        // Fast path: there is only one element in the antichain.
+                        // All times in `proposals` must be greater or equal to it.
+                        if antichain.len() == 1 && !antichain.less_equal(capability.time()) {
+                            stash_additions
+                                .entry(capability.delayed(&antichain[0]))
+                                .or_insert(Vec::new())
+                                .extend(proposals.drain(..));
+                        }
+                        else if antichain.len() > 1 {
+                            // Any remaining times should peel off elements from `proposals`.
+                            let mut additions = vec![Vec::new(); antichain.len()];
+                            for (data, initial, diff) in proposals.drain(..) {
+                                use timely::PartialOrder;
+                                let position = antichain.iter().position(|t| t.less_equal(&initial)).unwrap();
+                                additions[position].push((data, initial, diff));
+                            }
+                            for (time, addition) in antichain.into_iter().zip(additions) {
+                                stash_additions
+                                    .entry(capability.delayed(&time))
+                                    .or_insert(Vec::new())
+                                    .extend(addition);
+                            }
+                        }
                     }
                 }
             }
@@ -242,6 +274,10 @@ where
 
             // drop fully processed capabilities.
             stash.retain(|_,proposals| !proposals.is_empty());
+            
+            for (capability, proposals) in stash_additions.into_iter() {
+                stash.entry(capability).or_insert(Vec::new()).extend(proposals);
+            }
 
             // The logical merging frontier depends on both input1 and stash.
             let mut frontier = timely::progress::frontier::Antichain::new();